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In this article we prove the explicit Mordell Conjecture for large familiesof curves. In addition, we introduce a method, of easy application, to computeall rational points on curves of quite general shape and increasing genus. Themethod bases on some explicit and sharp estimates for the height of suchrational points, and the bounds are small enough to successfully implement acomputer search. As an evidence of the simplicity of its application, wepresent a variety of explicit examples and explain how to produce many others.In the appendix our method is compared in detail to the classical method ofManin-Demjanenko and the analysis of our explicit examples is carried toconclusion.

The explicit Mordell Conjecture for families of curves (with an appendix by M. Stoll)